scholarly journals Green‘s function approach in approximate controllability problems.

2016 ◽  
Vol 69 (2) ◽  
pp. 3-22 ◽  
Author(s):  
A.S. Avetisyan ◽  
As.Zh. Khurshudyan
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Approximate Controllability ◽  
Function Approach ◽  
Controllability Problems

Green’s function approach in approximate controllability for nonlinear physical processes

2017 ◽  
Vol 32 (21) ◽  
pp. 1730015 ◽  
Author(s):  
Ara S. Avetisyan ◽  
Asatur Zh. Khurshudyan
Keyword(s):  
Dynamical Systems ◽  
Green’S Function ◽  
Green's Function ◽  
Approximate Controllability ◽  
Nonlinear Dynamical Systems ◽  
Initial Boundary ◽  
Function Approach ◽  
Initial Boundary Value Problems ◽  
Nonlinear Dynamical ◽  
Initial Boundary Value

Recently, Green’s function approach is used to approximate (initial) boundary value problems for nonlinear dynamical systems. Here, we use this approach for derivation of a simple criterion for approximate controllability of nonlinear dynamical systems. Simple and easy-to-check criteria on system parameters are derived for process controllability. Examples from existing references are considered.

Resolving controls for exact and approximate controllability of viscous Burgers’ equation: The Green’s function approach

2018 ◽  
Vol 29 (06) ◽  
pp. 1850045 ◽  
Author(s):  
Asatur Zh. Khurshudyan
Keyword(s):  
Heat Equation ◽  
Green’S Function ◽  
Green's Function ◽  
Approximate Controllability ◽  
Burgers Equation ◽  
Sufficient Conditions ◽  
Exact Controllability ◽  
Function Approach ◽  
Dimensional System ◽  
Linear Algebraic Equations

In this paper, we consider a nonlinear control problem for one-dimensional viscous Burgers’ equation associated with a controlled linear heat equation by means of the Hopf–Cole transformation. The control is carried out by the time-dependent intensity of a distributed heat source influencing the heat equation. The set of admissible controls consists of compactly supported [Formula: see text] functions. Using the Green’s function approach, we analyze the possibilities of exact and approximate establishment of a given terminal state for the associated nonlinear Burgers’ equation within a desired amount of time. It is shown that the exact controllability of the associated Burgers’ equation and the heat equation are equivalent. Furthermore, sufficient conditions for the approximate controllability are derived. The set of resolving controls is constructed in both cases. The determination of the resolving controls providing exact controllability is reduced to an infinite-dimensional system of linear algebraic equations. By means of the heuristic method of resolving control determination, parametric hierarchies of solutions providing approximate controllability are constructed. The results of a numerical simulation supporting the theoretical derivations are discussed.

scholarly journals Controlling Power Consumption in a Heterogeneous Population Model of TCLs with Diffusion: The Green’s Function Approach

Mathematics ◽  
2019 ◽  
Vol 7 (6) ◽  
pp. 523
Author(s):  
Md Musabbir Hossain ◽  
Asatur Zh. Khurshudyan
Keyword(s):  
Power Consumption ◽  
Green’S Function ◽  
Green's Function ◽  
Approximate Controllability ◽  
Operation Time ◽  
Threshold Value ◽  
Heterogeneous Population ◽  
Point Of View ◽  
Function Approach ◽  
Initial Temperature Distribution

We consider a control problem for a diffusive PDE model of heterogeneous population of thermostatically controlled loads (TCLs) aiming to balance the aggregate power consumption within a given amount of time. Using the Green’s function approach, the problem is formulated as an approximate controllability problem for a residue depending on control parameters nonlinearly. A sufficient condition for approximate controllability is derived in terms of initial temperature distribution, operation time of TCLs and threshold value of the aggregate power consumption. Case studies allow to reveal the advantages of the proposed solution from numerical calculations point of view.

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